Journal of Pure and Applied Algebra 208 (2007) 39–52www.elsevier.com/locate/jpaa

Artin–Tits groups with CAT(0) Deligne complex

Eddy Godelle

Universite de Caen, Laboratoire de Mathematiques Nicolas Oresme, CNRS UMR 6139, 14032 Caen cedex, France

Received 9 September 2004; received in revised form 4 October 2005Available online 18 January 2006

Communicated by C. Kassel

Abstract

Let (A, S) be an Artin–Tits system, and AX be the standard parabolic subgroup of A generated by a subset X of S. Under thehypothesis that the Deligne complex has a CAT(0) geometric realization, we prove that the normalizer and the commensurator ofAX in A are equal. Furthermore, if AX is of spherical type, these subgroups are the product of AX with the quasi-centralizer of AXin A. For two-dimensional Artin–Tits groups, the result still holds without any sphericality hypothesis on X . We explicitly describethe elements of this quasi-centralizer.c© 2005 Elsevier B.V. All rights reserved.

MSC: 20F36; 57M07; 53C23

Introduction

Artin–Tits groups are a natural generalization of braid groups. They are defined by presentations involving relationssimilar to the standard braid relations but with length not necessarily equal to 2 or 3. The properties of generalArtin–Tits groups remain mysterious although some special families, like the family of spherical type Artin–Titsgroups (see Section 1 for a definition), are better known. An Artin–Tits group has a natural family of subgroups,namely the so-called parabolic subgroups: a standard parabolic subgroup is a subgroup generated by a subset of thedistinguished generating set; a parabolic subgroup is a subgroup that is conjugated to a standard parabolic subgroup.A standard parabolic subgroup is itself (and canonically) an Artin–Tits group [16]. It turns out that the study of thefamily of parabolic subgroups is crucial for understanding the whole group. Significant results have been proved byVan der Lek in [16] but a lot of questions remain open.

In the last few years larger families of Artin–Tits groups have been studied and are now well understood, especiallythe family of spherical type Artin–Tits groups (see [9,10,13,15]) and the family of FC type Artin–Tits groups(see [11]). In these cases, both combinatorial and geometrical methods have been successfully applied. Here weaddress three specific properties of Artin–Tits groups involving the normalizer, the parabolic subgroups and thecategory of ribbons. These properties were known to hold in some special cases, and we conjecture they alwaysdo. The aim of this paper is to establish the three properties for new cases.

The first property is concerned with the relation between an Artin–Tits group and its parabolic subgroups.We denote by ComAS (AX ), NAS (AX ) and QZAS

(X) the commensurator, the normalizer and the quasi-centralizer,

E-mail address: eddy.godelle@math.unicaen.fr.

0022-4049/$ - see front matter c© 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jpaa.2005.11.012

brought to you by COREView metadata, citation and similar papers at core.ac.uk

provided by Elsevier - Publisher Connector

40 E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52

respectively, of the standard parabolic subgroup AX in the Artin–Tits group AS (see the next section for a precisedefinition).

Definition 1. Let (AS, S) be an Artin–Tits system, and X be a subset of S. We say that the standard parabolic subgroupAX has Property (?) when

ComAS (AX ) = NAS (AX ) = AX · QZAS(X). (?)

We say that the group AS has Property (?) if all its standard parabolic subgroups have Property (?).

Some inclusions in (?) are obvious and Property (?) says that the commensurator (and thereby the normalizer) of aparabolic subgroup is small.

The second property describes the inclusion of a parabolic subgroup in another.

Definition 2. Let (AS, S) be an Artin–Tits system, and X, Y be included in S. We say that the pair (X, Y ) has Property(??) if for every g in AS , we have the implication

g AX g−1⊆ AY ⇒ ∃h ∈ AY ∃Z ⊆ Y such that g AX g−1

= h AZ h−1. (??)

We say that the group AS has Property (??) if all pairs (X, Y ) with X, Y included in S have Property (??).

In other words, if AS has Property (??), then every parabolic subgroup of AS included in a standard parabolic subgroupAY is a parabolic subgroup of AY .

The third property is concerned with the problem of conjugacy between two parabolic subgroups and it involvesthe category of ribbons. In the case of the braid group, there exists a natural notion of ribbon associated with apair of strings that move together along the braid. The notion was used in [9] to describe the quasi-centralizersand subsequently generalized from a combinatorial viewpoint in [13] and [10]. Roughly speaking, the objects ofthe category of conjugators Conj(S) are parabolic subgroups of AS and the morphisms are the elements of ASthat conjugate a standard parabolic subgroup to another. The subcategory of ribbons Ribb(S) has the same objectsas Conj(S), but the morphisms are only the ones that can be constructed as ribbons—see the next section for aprecise definition of Conj(S) and Ribb(S). Denote by Conj(S; X, Y ) and Ribb(S; X, Y ) the set of morphisms fromthe subgroup AX to AY in the categories Conj(S) and Ribb(S) respectively.

Definition 3. Let (AS, S) be an Artin–Tits system, and X, Y be two subsets of S. We say that the pair (X, Y ) hasProperty (? ? ?) if Conj(S; X, Y ) and Ribb(S; X, Y ) coincide. We say that the group AS has Property (? ? ?) if

Conj(S) = Ribb(S). (? ? ?)

In other words, the group AS has Property (? ? ?) when all pairs (X, Y ) have Property (? ? ?).

It is known that spherical type Artin–Tits groups and Artin–Tits groups of FC type have Properties (?), (??) and(? ? ?) [10,11].

Conjecture 1. Every Artin–Tits group has Properties (?), (??) and (? ? ?).

The Intuition that Conjecture 1 holds is supported by the fact that its monoid counterpart is true (see [10]).The aim of this paper is to prove Properties (?), (??) and (? ? ?) for some Artin–Tits groups, among which are

the two-dimensional Artin–Tits groups, that are neither of FC type nor of spherical type (we recall that an Artin–Titsgroup AS is two-dimensional if for every subset X of S with cardinality at least 3, the standard parabolic subgroupAX is not of spherical type).

Our methods are geometric, and two of our main tools are the Deligne complex, introduced in [8] and generalizedin [7], and the CAT(0) theory.

Let us postpone most of the definitions to the next sections and just state the results precisely. We denote by DS theDeligne complex of AS . If X is a subset of S, we denote by AX the standard parabolic subgroup of AS generated byX . The results we prove are as follows (see Sections 2 and 4.3. for the definition of the Deligne complex and CAT(0)

spaces):

E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52 41

Theorem 1. Assume that (AS, S) is an Artin–Tits system such that the Deligne complex DS of AS has a piecewiseEuclidean CAT(0) geometric realization ΓS . Let X be a subset of S such that AX is of spherical type; then

(i) The subgroup AX has Property (?).(ii) Let Y be in S; assume that AY is of spherical type, or more generally that the geometric subcomplex ΓY of ΓS

associated with Y is convex. Then, the pair (X, Y ) has Property (??).(iii) For every subset Y of S such that the geometric subcomplex ΓY of ΓS associated with Y is convex, the pair

(X, Y ) has Property (? ? ?).

Theorem 2. Assume that (AS, S) is an Artin–Tits system such that the Deligne complex DS of AS has a piecewiseEuclidean CAT(0) geometric realization ΓS . Let X be a subset of S such that the geometric subcomplex ΓX of ΓSassociated with X is convex; then we have

ComAS (AX ) = NAS (AX ).

Theorem 3. Assume that (AS, S) is a two-dimensional Artin–Tits system. Then the group AS has Properties (?), (??)and (? ? ?).

Charney proved in [5] that, if the chosen CAT(0) realization of DS is the Moussong one (see Section 2), then ΓY isconvex for every subset Y of S. She proved that ΓY is also convex for the cubical metric on the Deligne complex forFC type Artin–Tits groups. As a consequence, Theorems 1 and 2 provide new proofs for the result that Artin–Titsgroups of FC type have Properties (?), (??) and (? ? ?). To the best of our knowledge, these results are new for thefamily of Artin–Tits groups such that the Moussong realization of the Deligne complex is CAT(0). Theorem 3 seemsto be related to no previous result.

The paper is organized as follows. In Section 1 we recall some properties of Artin–Tits groups. In Section 2 weintroduce our geometric tools. Section 3 is devoted to the case of parabolic subgroups of spherical type. In particularwe prove Theorems 1 and 2 (Corollaries 3.5, 3.7 and 3.11). In Section 4 we begin with the study of non-sphericaltype parabolic subgroups of every Artin–Tits group and, after investigating geometric properties of CAT(0) spaces,we address the case of two-dimensional Artin–Tits groups and prove Theorem 3.

1. Artin–Tits groups

We recall here some basic definitions, notation and results on Artin–Tits groups. Let S be a finite set andM = (ms,t )s,t∈S be a symmetric matrix with ms,s = 1 for s in S and ms,t in {2, 3, 4, . . .} ∪ {∞} for s 6= t in S.The Artin–Tits system associated with M is the pair (AS, S) where AS is the group defined by the presentation

AS = 〈S | sts · · ·︸ ︷︷ ︸ms,t terms

= tst · · ·︸ ︷︷ ︸ms,t terms

; ∀s, t ∈ S, s 6= t and ms,t 6= ∞〉. (∗)

The relations sts · · ·︸ ︷︷ ︸ms,t terms

= tst · · ·︸ ︷︷ ︸ms,t terms

are called braid relations and the group AS is said to be an Artin–Tits group. For

instance, if S = {s1, . . . , sn} with msi ,s j = 3 for |i − j | = 1 and msi ,s j = 2 otherwise, then the associated Artin–Titsgroup is the braid group on n + 1 strings. We denote by A+

S the submonoid of AS generated by S. This monoid A+

Shas the same presentation as the group AS , considered as a monoid presentation [14] and is cancellative. When weadd the relations s2

= 1 to the presentation (∗), we obtain the Coxeter group WS associated with AS . One says thatAS is of spherical type if WS is finite. The matrix M may be represented by a graph, whose vertex set is S and wherean edge connects two vertices if ms,t ≥ 3; these edges are labelled by ms,t for ms,t ≥ 4. One says that AS (or simplyS) is indecomposable when this graph is connected. The indecomposable components of S are the maximal subsetsof S that are indecomposable.

A subgroup AX of AS generated by a subset X of S is called a standard parabolic subgroup, and a subgroup ofAS conjugated to a standard parabolic subgroup is called a parabolic subgroup. Van der Lek has shown in [16] that(AX , X) is canonically isomorphic to the Artin–Tits system associated with the matrix (ms,t )s,t∈X ; its graph is thesubgraph induced by X .

42 E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52

For every two elements a and b in the monoid A+

S , one says that a left-divides b if b = ac for some c in A+

S ; inthat case, we write a ≺ b. We define in the same way the right-divisibility and write b � a if a right-divides b. Thefollowing result is well known:

Lemma 1.1 ([4]). Let (AS, S) be an Artin–Tits system. Then, the set S has a least common multiple (lcm) in A+

S forleft-divisibility if and only if S has a least common multiple (lcm) in A+

S for right-divisibility if and only if AS is ofspherical type. In that case, these two lcm’s are equal and are denoted by ∆S .

As a consequence, for every spherical type Artin–Tits group AS and for every subset X of S, the subgroup AX isof spherical type and ∆X left-divides ∆S in A+

S .We recall now some notation introduced in [11] and define the categories Conj(S) and Ribb(S).

Notation 1.2. Let (AS, S) be an Artin–Tits system and X be a subset of S.

(i) We denote by Xs and Xas respectively the union of the spherical type indecomposable components of X and theunion of the non-spherical type indecomposable components of X .

(ii) We set X⊥= {s ∈ S; ∀t ∈ X, ms,t = 2} and for k in N, X k

= {sk; s ∈ X}; in particular ∅

⊥= S.

Note that we always have X ∩ X⊥= ∅ because ms,s = 1 for every s in S. For short we will write X⊥

as for (Xas)⊥;

this notation is unambiguous in our context.The following definitions of the categories Conj(S) and Ribb(S) are technical. The reader may prefer to keep in

mind the rough definition of the introduction and skip the precise definition.

Definition 1.3. Let (AS, S) be an Artin–Tits system.

(i) We define the groupoid Conj(S) as follows: The objects of Conj(S) are all the subsets of S and the setConj(S; X, Y ) of morphisms from X to Y is in 1–1 correspondence with the set {g ∈ AS | gXg−1

= Y }.The composition of morphisms is defined by the product in AS : g ◦ f = g f .

(ii) Let X, Y be two subsets of S; we say that an element w of Conj(S; X, Y ) is a positive elementary Y -ribbon-X if either w = ∆Z holds for some indecomposable component Z of X or there exists t ∈ S such that theindecomposable component Z of X ∪ {t} containing t is of spherical type and w = ∆Z∆−1

Z−{t}.We say that an element w of Conj(S; X, Y ) is an elementary Y -ribbon-X if it is a positive elementary ribbon

or w−1 is a positive elementary X -ribbon-Y .(iii) We denote by Ribb(S) the smallest subcategory of Conj(S) that has the same objects as Conj(S) and that contains

the elementary ribbons; the set of morphisms from X to Y in Ribb(S) is denoted as Ribb(S; X, Y ) and its elementsare called Y -ribbons-X .

Recall that for X a subset of S, The quasi-centralizer QZAS(X) of the subgroup AX in AS is

QZAS(X) = {g ∈ AS|gX = Xg}.

Hence, by definition, we have QZAS(X) = Conj(S; X, X).

Proposition 1.4 ([10] Proposition 2.1). Let (AS, S) be an Artin–Tits system of spherical type and X, Y be two subsetsof S. Let k be in Z − {0} and g be in AS; then the following statements are equivalent:(1) g AX g−1

⊆ AY ;(2) g∆k

X g−1∈ AY ;

(3) g = yx for some y ∈ AY , x ∈ Conj(S; R, X) for some R ⊆ Y .

Let us finish this section with the definition of two families of Artin–Tits groups, namely the family of two-dimensional Artin–Tits groups and the family of FC type Artin–Tits groups. We do not study the latter family in thispaper, but we refer to it several times, and that is why we give its definition for completeness.

Definition 1.5. Let (AS, S) be an Artin–Tits system. Following [5], one says that (AS, S) is a two-dimensionalArtin–Tits system (or that AS is a two-dimensional Artin–Tits group) if for every subset X of S with cardinalityat least 3, AX is not of spherical type.

Definition 1.6. Let (AS, S) be an Artin–Tits system. One says that (AS, S) is an Artin–Tits system of FC type if thefollowing property holds: if X is a subset of S such that for all s, t in X , ms,t is finite, then AX is of spherical type.

E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52 43

2. Deligne complex and CAT(0) realization

In this section, we introduce one of our main tools : the Deligne complex, which is a simplicial complex on whichthe Artin–Tits group acts. In the first subsection, we define the Deligne complex. In the second subsection, we recallthe notion of a CAT(0) space. In the third subsection, we recall the construction of a particular realization of theDeligne complex, namely the Moussong realization, which is known to be CAT(0) in the case of two-dimensionalArtin–Tits groups and conjecturally CAT(0) for every Artin–Tits group.

2.1. The Deligne complex

We are now going to introduce the Deligne complex and some of its geometric realizations. The Deligne complexwas initially defined in [8] by Deligne in the case of spherical type Artin–Tits groups. The construction was generalizedby Charney and Davis in [7]. Let (AS, S) be an Artin–Tits group; we set

S f,S = {T ⊆ S; AT is of spherical type}

and

ASS f,S = {x AT ; x ∈ AS and T ∈ S f,S}.

Recall that the complex associated with a partially ordered set (P, ≤) is the abstract simplicial complex whose verticesare the elements of P and where a finite set of vertices spans a simplex if these vertices can be ordered into anincreasing sequence, for the partial order ≤, of elements of P . The Deligne complex DS is the complex associatedwith ASS f,S partially ordered by inclusion. Note that x AX is a subset of y AY if and only if X is a subset of Y andy−1x is in AY . In order to prevent confusion between the vertex AX and the standard parabolic subgroup AX , wesometimes write eAX for the vertex, where e is the unit element of AS .

The group AS acts by left multiplication on DS and thus simplicially; the subcomplex KS of DS is the subcomplexof DS generated by the vertices eAT where T is in S f,S ; it is a fundamental domain of DS for the action of AS andis the union of the K X for X in S f,S . An interval of DS is called an abstract cell; if K is an abstract cell of DS , thenit has a greatest vertex a AX and a lowest vertex a AY with X, Y in S f,S and Y a subset of X ; the set of vertices ofK is {a AZ ; Y ⊆ Z ⊆ X} and its dimension is #(X − Y ). In that case, we write K = K (a AY , a AX ). For instance,K X = K (A∅, AX ).

One can associate a geometric realization with the abstract simplicial complex DS . In that case, the geometricrealization of an abstract cell is called a cell. We will only consider realizations such that AS acts by isometries: wechoose first a realization of KS , and then we extend it to DS using the action of AS . The complex DS is commonlyidentified with a chosen realization, even if such a realization is not unique.

If D is a set and G a group that acts on D, then the subgroups Fix(C) = {g ∈ G; ∀x ∈ C, g · x = x} andStab(C) = {g ∈ G; g · C = C} are called the pointwise stabilizer and the stabilizer, respectively, of the subset Cof D. If C = {x}, we write Fix(x) for Fix({x}) = Stab({x}). The stabilizer of a vertex a AX of DS is a AX a−1 andthe stabilizer of cell K = K (a AY , a AX ) is a AY a−1; it is also its pointwise stabilizer. Let ΓS be a realization of DS .Since AS acts by isometries and simplicially on ΓS , the pointwise stabilizer of a point x is a AY a−1 if x is a vertexof the form x = a AY or if x is in the interior of a cell K = K (a AY , a AX ). If K1 and K2 are two cells of DS thenwe denote by span(K1, K2) the smallest cell of DS that contains K1 and K2, when it exists. Let us recall the twofollowing results:

Lemma 2.1 ([16] Theorem 4.13). Let (AS, S) be an Artin–Tits system and X, Y be subsets of S. Then, AX ∩ AY =

AX∩Y .

Lemma 2.2 ([1] Lemma 4.1). Let (AS, S) be an Artin–Tits system. Let K1 = K (a1 AR1 , a1 AT1) and K2 =

K (a2 AR2 , a2 AT2) be two cells of the complex DS . Then, span(K1, K2) exists if and only if T1 ∪ T2 ∈ S f,Sand a1 AR1 ∩ a2 AR2 6= ∅. Furthermore, in that case, we have span(K1, K2) = K (bAR1∩R2 , bAT1∪T2) with b ina1 AR1 ∩ a2 AR2 .

In Lemma 4.1 of [1] the Artin–Tits group is assumed to be of type FC, but the statement remains true in the generalcontext, and the proof is the same.

44 E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52

2.2. CAT(0) spaces

Since the seminal work of M. Gromov, CAT(0) spaces have become an important tool in group theory. Theexistence for a group of a CAT(0) space on which the group acts by isometries implies several properties for thatgroup. In this section, we recall basic definitions and results on CAT(0) spaces. We refer the reader to [3] for moredetails. Let (X, d) be a metric space. A geodesic between two points x and y of X is an isometry γ from [0; d(x, y)]

to X such that γ {0; 1} = {x; y}. One says that this geodesic is oriented from x to y if γ (0) = x . Following [3], weidentify γ with its image, which is denoted by [x, y] (even though the geodesic is not unique). A geodesic triangle∆(x, y, z) of X is the union of three geodesics [x, y], [y, z] and [z, x] of X , and a comparison triangle is a triangle∆(x, y, z) of the Euclidean plane E2 such that dE2(a, b) = d(a, b) for every a, b of {x, y, z}. If p is on [x, y], thecomparison point of p in ∆(x, y, z) is the point p of [x, y] such that dE2(x, p) = d(x, p). One says that the triangle∆(x, y, z) verifies the CAT(0) hypothesis if for every p, q in ∆(x, y, z) one has d(p, q) ≤ dE2(p, q). The metricspace X is said to be CAT(0) if every two points of X can be joined by a geodesic and every geodesic triangle of Xverifies the CAT(0) hypothesis.

When a metric space is CAT(0), then numerous properties of the Euclidean spaces still hold; let us start with the keyone, which follows easily from the definition (see [3] for instance):

Proposition 2.3. If (X, d) is a CAT(0) metric space, then there exists a unique geodesic (up to orientation) betweenevery two points of X.

If (X, d) is a metric space and Y is a non-empty subspace of X , then Y is said to be convex if for every two points x ,y in Y , every geodesic [x, y] is included in Y .

If X is CAT(0), one can define the angle 6 x (y, z) between two geodesics [x, y] and [x, z]; this angle is called theAlexandrov angle of the two geodesics. We have the following properties:

Proposition 2.4 ([3] Proposition II 2.4). Let (X, d) be a CAT(0) metric space and C a complete non-empty convexsubspace of X; then for every point x of X, there exists a unique point πC (x) of C such that

d(x, πC (x)) = Inf{d(x, y) | y ∈ C}.

Furthermore,

(i) if x ′∈ [x, πC (x)] then πC (x) = πC (x ′);

(ii) if x 6∈ C and y ∈ C then 6πC (x)(x, y) ≥

π2 ;

(iii) the map x 7→ πC (x) is a non-decreasing retraction.

Proposition 2.5 ([3] II.2.11 The Flat Quadrilateral Theorem). Consider four distinct points p, q, r and s in a CAT(0)

space X. Let α = 6 p(q, s), β = 6 q(p, r), γ = 6 r (q, s) and δ = 6 s(r, p). If α + β + γ + δ ≥ 2π , then this sum isequal to 2π and the convex hull [p, q, r, s] of the four points p, q, r and s is isometric to the convex hull of a convexquadrilateral in E2.

2.3. The Moussong realization

We describe now a particular realization of the Deligne complex constructed by Charney and Davis in [7]. Wefollow the description given in [5]. The so-called realization of the Deligne complex was introduced by Moussong inhis thesis [12].

E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52 45

Let T be in S f,S ; consider a real vector space ET of dimension |T |, and the standard realization of WT as an orthogonalreflection group on ET (see [2] Chapter V). Let CT be the closed cone that is the fundamental domain for the actionof WT on ET and let x∅ be the unique point of CT that is at distance 1 from every wall of CT . Consider XT the convexhull of the WT -orbit of x∅; it is a cell and it and the convex hull F∗

R of the WR-orbit of x∅ for R ⊆ T are faces of XT .For s in T , denote by Fs the wall of ET fixed by the reflection s and for R ⊆ T set FR = ∩s∈R Fs . Then FR andF∗

R are orthogonal and intersect in a single point xR . The intersection of XT and CT is combinatorially a cube withvertices xR for R ⊆ T . The Euclidean realization of KS is obtained by choosing XT ∩CT for realization of KT (wherexR is the realization of AR) for every T in S f,S ; this is consistent because for R ⊆ T , the face F∗

R is isometric to X R .We extend the cellular piecewise Euclidean structure of KS to DS using the action of AS . This geometric realizationis called the Moussong realization of the Deligne complex. In [7] Charney and Davis stated the following conjecture:

Conjecture 2.6 ([7] Conjecture 4.4.4). The Moussong realization of every Artin–Tits group is CAT(0).

They proved this conjecture for the family of the two-dimensional Artin–Tits groups:

Theorem 2.7. Let (AS, S) be a two-dimensional Artin–Tits system; then the Moussong realization of AS is CAT(0).

The Moussong realization is not the only interesting realization for an Artin–Tits group. For instance the Delignecomplex of an Artin–Tits group of type FC has a realization that is CAT(0) ([7] Theorem 4.3.5). Some other cases areknown (see [6] Corollary 5.5)

In the last section, we will use the following property:

Proposition 2.8 ([5] Lemma 5.1). Let (AS, S) be an Artin–Tits system and T be a subset of S. Denote by ΓS and ΓTthe Moussong realizations of DS and DT respectively; we consider DT (resp. ΓT ) as a subcomplex of DS (resp. ΓS).Assume that ΓS is CAT(0); then ΓT is a (closed) convex subspace of ΓS and it is CAT(0).

3. Spherical type parabolic subgroups

The following result is the key point of our study.

Theorem 3.1. Let (AS, S) be an Artin–Tits system; let X, Y be two subsets of S of spherical type, let g be in ASand k be in Z − {0}. If DS has a (piecewise Euclidean simplicial) geometric realization ΓS that is CAT(0) then thefollowing statements are equivalent:

(1) g AX g−1⊆ AY ;

(2) g∆kX g−1

∈ AY ;(3) g ∈ AY · Ribb(S; X, R) for some R ⊆ Y .

In order to prove this theorem, let us introduce the following definitions:

46 E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52

Definition 3.2. Let (X, d) be a CAT(0) metric space. Let C be a non-empty convex subset of X and [x, y] a geodesicof X . We say that [x, y] crosses C transversally if C ∩ [x, y] is a singleton and we say that [x, y] crosses C internallyif [x, y] ∩ C contains at least two distinct points.

Definition 3.3. Let (AS, S) be an Artin–Tits system and ΓS a realization of DS . If x, y are two points of ΓS , wedenote by span(x) and span(x, y) respectively the smallest closed cell that contains x and the smallest closed cell thatcontains both x and y, if it exists. If x, y are two points of ΓS and z is in [x, y], we say that z is a transversal point of[x, y] when [x, y] crosses span(z) transversally.

Note that this definition is consistent with the notation span(K1, K2) defined in Section 2.1 above.The following proposition is an immediate consequence of Corollary 7.29 page 110 of [3].

Proposition 3.4. Let x, y be two points of a CAT(0) realization ΓS of DS and γ : [0, d(x, y)] → E be the geodesicfrom x to y. There exists a subdivision 0 = t0 < t1 < · · · < tn = d(x, y) such that for every i ∈ {1, . . . , n − 1},ci = γ (ti ) is a transversal point of [x, y] and for every i ∈ {0, . . . , n−1}, the geodesic γ crosses span(γ (ti ), γ (ti+1))

internally. Furthermore, using convexity, one has

span(ci , ci+1) ∩ [x, y] = [ci , ci+1].

Proof of Theorem 3.1. It is clear that (3) ⇒ (1) ⇒ (2), so it is enough to prove (2) ⇒ (3). Recall that byProposition 1.4 we know the implication is true when AS is of spherical type. Assume g∆k

X g−1∈ AY with

k ∈ Z − {0}. It follows that the vertex g−1 AY of ΓS is fixed by ∆kX . Let c be a point of ΓS and consider the

geodesic γ from e · AX to c in ΓS . Set span(c) = K (h AZ , h AU ); with this notation, the pointwise stabilizer of cis h AZ h−1. Consider the subdivision 0 = t0 < t1 < · · · < tn = d(AX , c) as in Proposition 3.4. Set ci = γ (ti )and span(ci ) = K (hi AZi , hi AUi ). The pointwise stabilizer of ci is then hi AZi h

−1i . Let us show, by induction on i ,

that if ci is fixed by ∆kX then h−1

i ∈ AZi · Ribb(S; X, Ri ) for some Ri ⊆ Zi . Applying this result to c = g−1 AY ,we will prove the theorem. Note that for i ≤ 1, the result follows from Proposition 1.3 because we are in a cell andthus in a spherical type parabolic subgroup; so assume i ≥ 2. Since both e · AX and c are fixed by ∆k

X , the isometry∆k

X fixes every point of the (unique) geodesic γ . Applying the induction hypothesis to ci−1, we get that h−1i−1 = uv

with u in AZi−1 and v in Ribb(S; X, Ri−1) for some subset Ri−1 of Zi−1. Now, by construction, span(ci−1, ci )

exists and is equal to K (αAZi−1∩Zi , αAUi−1∪Ui ) with α ∈ hi−1 AZi−1 ∩ hi AZi . Hence h−1i hi−1u is in AUi−1∪Ui and

(h−1i hi−1u)∆k

Ri(h−1

i hi−1u)−1 is equal to h−1i ∆k

X hi and therefore is in AZi by assumption. Applying Proposition 1.4

in AUi−1∪Ui , we get that h−1i hi−1u = u′v′ with u′

∈ AZi and v′∈ Ribb(S; Ri−1, Ri ) for some Ri ⊆ Zi . Thus

hi = u′v′v and we are done since v′v ∈ Ribb(S; Ri−1, Ri )Ribb(S; X, Ri−1) is included in Ribb(S; X, Ri ). �

The two main families for which CAT(0) realizations of the Deligne complexes are known are FC type Artin–Titsgroups and two-dimensional Artin–Tits groups. For both of them, every Deligne subcomplex of a parabolic subgroupis convex; hence the following corollaries apply in both cases. Note that, for FC type Artin–Tits groups these resultsare already known, and have been proved by a non-geometric approach (see [11]). In the case of the braid group, thefirst part of Corollary 3.8 has been proved in [9].

Recall that the normalizer NAS (AX ) of a standard parabolic subgroup AX in AS is the subgroup of AS defined by

NAS (AX ) = {g ∈ AS|gX ⊆ AX g}

and that an element g in AS is in the commensurator ComAS (AX ) of AX when g AX g−1∩ AX has a finite index

both in AX and in g AX g−1. Note that ComAS (AX ) is a subgroup of AS . In order to state the next result, we needsome notation: if G is a group and G1, G2 are two subgroups such that G2 normalizes G1, then we denote byG1 · G2 the subgroup generated by G1 and G2. This subgroup is both {g1g2 ∈ G | g1 ∈ G1; g2 ∈ G2} and{g2g1 ∈ G | g1 ∈ G1; g2 ∈ G2}.

Corollary 3.5. Let (AS, S) be an Artin–Tits system; let X, Y be two subsets of S of spherical type. If DS has a(piecewise Euclidean simplicial) geometric realization ΓS that is CAT(0), then

(i) Conj(S; X, Y ) = Ribb(S; X, Y ) and QZAS(X) = Ribb(S; X, X);

(ii) ComAS (AX ) = NAS (AX ) = AX · QZAS(X).

E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52 47

Proof. (i) By definition and Theorem 3.1 we have a sequence of inclusions

Ribb(S; X, Y ) ⊆ Conj(S; X, Y ) ⊆ AY · Ribb(S; X, Y ).

It follows that Conj(S; X, Y ) = Conj(Y ; Y, Y ) · Ribb(S; X, Y ). But, we also have Conj(Y ; Y, Y ) =

Ribb(Y ; Y, Y ) = QZAY(Y ) since AY is of spherical type. Thus Ribb(S; X, Y ) = Conj(S; X, Y ).

(ii) It is clear using definition that ComAS (AX ) ⊃ NAS (AX ) ⊃ AX · QZAS(X). Conversely, let g be in ComAS (AX ).

Then, the set of left-cosets ∆ jX (AX ∩ g−1 AX g) with j in N − {0} in AX is finite. Then, there exists k in N − {0}

such that g∆kX g−1 is in AX and g is in AX · QZAS

(X). �

The following lemma will be crucial when proving Corollaries 3.7 and 3.10 and in the proof of Theorem 4.8.

Lemma 3.6. Let (D, d) be a CAT(0) metric space and G a group that acts by isometries on D. Let C be a non-emptyconvex subspace of D and x a point of D; then Fix(x) ∩ Stab(C) ⊆ Fix(πC (x)).

Proof. One can assume that Fix(x) ∩ Stab(C) 6= {0}. Let g be in Fix(x) ∩ Stab(C); since g ∈ Stab(C), the pointg ·πC (x) is in C . Furthermore, G acts by isometries, then d(x, πC (x)) = d(g · x, g ·πC (x)) = d(x, g ·πC (x)). Thesetwo properties and the uniqueness part of Proposition 2.4 imply that g · πC (x) = πC (x). �

Corollary 3.7. If in the statement of Theorem 3.1 we replace the hypothesis “Y is of spherical type” by “ΓY isconvex in ΓS and AY of non-spherical type” then the conclusion remains true. Furthermore, the subset R from (3) ofTheorem 3.1 is of spherical type and Conj(S; X, Y ) = Ribb(S; X, Y ) = ∅.

Proof. (i) As for Theorem 3.1, the implications (3) ⇒ (1) ⇒ (2) are clear. So assume (2) and let us show (3). Thekey idea is to prove that g∆k

X g−1 is in a spherical type parabolic subgroup of AY and then to apply Theorem 3.1to conclude. Let p = πg−1·ΓY

(AX ) be the projection of the vertex x = eAX of ΓS on the convex subspaceg−1

· ΓY . As explained in Section 2.1, there exist h ∈ AY and Z ∈ S f,Y such that the pointwise stabilizer of pis g−1h AZ (g−1h)−1. By Lemma 3.6 we have Fix(x) ∩ Stab(g−1

· ΓY ) ⊆ Fix(p). In particular, ∆kX · p = p, that

is ∆kX is in Fix(p) = g−1h AZ (g−1h)−1. By (2)⇒(3) of Theorem 3.1, we get that g = huv with u in AZ and v in

Ribb(S; X, R) for some subset R of Z . Finally, since R is a subset of Z and Z is of spherical type, we get that R is ofspherical type. The assertion Conj(S; X, Y ) = ∅ is consequently obvious. �

Corollary 3.8. Let (AS, S) be an Artin–Tits system and assume that DS has a (piecewise Euclidean simplicial)geometric realization ΓS that is CAT(0).

(i) Let s, t be in S and g be in AS , then gsng−1= tn for some n in Z − {0} if and only if gsg−1

= t .(ii) Let s be in S, g be in AS and Y be a subset of S such that ΓY is convex in ΓS . Then, gsng−1

∈ AY for some n inZ − {0} if and only if gsg−1 is in AY .

Proof. Point (i) is a special case of (ii). In (ii), the “if” part is clear, and the “only if” part is a consequence of(2) ⇐⇒ (3) of Theorem 3.1 (and Corollary 3.7) with X = {s}, k = n and k = 1. �

Lemma 3.9. Let (AS, S) be an Artin–Tits system and ΓS a CAT(0) realization of DS . Let x be a point of ΓS , g in ASand Y a subset of S such that ΓY is convex and x is not in g · ΓY . Then the endpoint πg·ΓY (x) is a transversal pointof the geodesic [x, πg·ΓY (x)].

Proof. Since span(πg·ΓY (x)) is a subset of g · ΓY and [x, πg·ΓY (x)] is a geodesic, [x, πg·ΓY (x)] intersects the interiorof span(πg·ΓY (x)) in a single point πg·ΓY (x). �

Corollary 3.10. Let (AS, S) be an Artin–Tits system; let X, Y be two subsets of S and g be in AS; assume that DShas a (piecewise Euclidean simplicial) geometric realization ΓS that is CAT(0) and that ΓY is convex in ΓS . Assumefurthermore that X is maximal in S such that AX is of spherical type. If g AX g−1 is a subgroup of AY then X is asubset of Y and g is in AY . If furthermore AY is of spherical type then X = Y and eAX is the unique point of ΓSfixed by the subgroup AX .

48 E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52

Proof. Assume eAX is not a vertex of g−1·ΓY . Consider the geodesic from the vertex x = eAX to p = πg−1·ΓY

(AX ).As in Corollary 3.7, and by Lemma 3.6, the endpoint p is fixed by AX . Since both extremities are fixed by AX , eachpoint of the geodesic [x, p] is fixed by AX . Let c be the first transversal point of the geodesic [x, p] distinct from x ;such a point exists by Lemma 3.9. Write Fix(c) = g1 AR and span(c) = K (g1 AR, g1 AT ). By the choice of c, thepoints e · AX and c span a cell, thus g1 AR = αAR with α ∈ AX and span(AX , c) = K (αAX∩R, αAX∪T ). But c isfixed by AX , hence AX ⊆ αARα−1. Since α is in AX , we get that X is a subset of R; this is impossible since bymaximality of X it follows that X = R and c = e · AX . It follows that the vertex e · AX is in g−1

· ΓY , that is X isincluded in Y and g is in AY . When AY is of spherical type, we get X = Y by maximality of X . �

Corollary 3.11. Let (AS, S) be an Artin–Tits system; Assume that the Deligne complex DS of AS has a piecewiseEuclidean geometric realization ΓS that is CAT(0). Let X, Y be two subsets of S such that ΓY is convex in ΓS . Then

(i) For every g in AS and every k ∈ Z − {0}, gX k g−1⊆ AY ⇐⇒ g AX g−1

⊆ AY where X k= {xk

| x ∈ X}.(ii) ComAS (AX ) = NAS (AX ).

Proof. Point (i) is an immediate consequence of Corollary 3.8(ii) applied to each element of X . In (ii), By definitionNAS (AX ) is included in ComAS (AX ). Conversely, let g be in ComAS (AX ). By the same argument as in the proof ofCorollary 3.8(ii), for each s of X there exists ks in N−{0} such that gsks g−1 is in AX . Hence there exists k in N−{0}

such that gX k g−1 is contained in AX ; by Part (i), it follows that g is in NAS (AX ). �

4. The case of non-spherical parabolic subgroups

This section is divided into three subsections. In the first subsection, we state Conjecture 4.2, which generalizesTheorem 3.1 by removing the restriction that AX is assumed to be of spherical type. Then, we show why, in orderto prove the equivalences in the property called (~) below, it is enough to prove them when the subgroup AX isan indecomposable non-spherical standard parabolic subgroup. In the second subsection, we establish geometricalproperties of CAT(0) spaces related to projection on convex subspaces. In the third subsection, using these geometricalproperties and a little extra trick, we prove the equivalence of Property (~)in the special case of a two-dimensionalArtin–Tits group. Hence we obtain a complete proof of Theorem 3.

4.1. The main conjecture

We refer the reader back to Section 1.5 for the definition of Xs and Xas .

Definition 4.1. Let (AS, S) be an Artin–Tits system. Let X, Y be two subsets of S. We say that the pair (X, Y ) hasproperty (~) when for every g in AS and every k in Z − {0}, the following statements are equivalent:

(1) g AX g−1⊆ AY ;

(2) g∆kXs

g−1∈ AY ; X k

as ⊆ AY and g = uv with u ∈ AY and v ∈ AX⊥as

;

(3) Xas ⊆ Y and g ∈ AY · Ribb(X⊥as; Xs, R) for some R ⊆ Y .

We say that the group AS has Property (~) when all pairs (X, Y ) have Property (~).

Theorem 3.1 suggests the following conjecture:

Conjecture 4.2. Let (AS, S) be an Artin–Tits system. The group AS has Property (~).

In [11] we prove by algebraic methods that Artin–Tits groups of FC type have Property (~) (with (2) slightly different).Note that (3) ⇒ (1). If we assume that DS has a CAT(0) realization and ΓY is convex, then we have (2) ⇒ (3) usingTheorem 3.1 and Corollary 3.8. Also (1) ⇒ g∆k

Xsg−1

∈ AY and gX kas g−1

⊆ AY . Hence the only thing to proveunder the hypothesis in Conjecture 4.2 is the implication

g AX g−1⊆ AY ⇒ g = uv with u ∈ AY and v ∈ AX⊥

as. (Ď)

This is because in that case X kas = vX k

asv−1

⊆ u−1 AY u = AY .

E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52 49

Proposition 4.3. Let (AS, S) be an Artin–Tits system such that the group AS has Property (~); then the group AShas Properties (?), (??) and (? ? ?).

Proof. Let (AS, S) be an Artin–Tits system and X a subset of S. Assume AS has Property (~). We haveComAS (AX ) = NAS (AX ) as in Corollary 3.11. Statement (1) ⇐⇒ (3) of Property (~) implies NAS (AX ) =

AX · QZAS(X). Properties (??) and (? ? ?) are also easy consequences of the equivalences of (1) and (3) in (~). �

We show now that the task of proving statement (Ď) for all subsets reduces to proving it in the case when X is aparticular type of standard parabolic subgroup.

Definition 4.4. Let F be a family of Artin–Tits systems. We say that F is an SBSPS-family if for every Artin–Titssystem (AS, S) of F and every subset X of S, the system (AX , X) is in F .

The notation “SBSPS” stands for “stable by standard parabolic subgroup”. For instance, the family of Artin–Titssystems of spherical type is an SBSPS-family, the family of Artin–Tits systems of FC type is an SBSPS-family andthe family of two-dimensional Artin–Tits systems is an SBSPS-family. Of course the family of all Artin–Tits systemsis an SBSPS-family. Finally, by Theorem 2.7 the family of Artin–Tits systems such that the Moussong realization oftheir Deligne complex is CAT(0) is an SBSPS-family.

Lemma 4.5 (First Restriction Lemma). Let F be an SBSPS-family. Assume that for every (AS, S) of F and every twosubsets X and Y of S such that AX is indecomposable, (Ď) is true. Then, (Ď) is true for every (AS, S) of F and everytwo subsets X and Y of S.

Proof. Assume (Ď) is true for every (AS, X, Y, g) such that (AS, S) is in F , g is in AS and X and Y are two subsetsof S such that AX is indecomposable. Let X be a subset of S and g ∈ AS such that g AX g−1

⊆ AY . Let us showthat (Ď) is true by induction on the number n of non-spherical indecomposable components of X . If n = 0 theresult is trivially true since Xas = ∅ and then AX⊥

as= AS . If n = 1, it is true by hypothesis applied to Xas since

g AXas g−1⊆ g AX g−1

⊆ AY . Let n ≥ 2. Assume the implication is true for every (AS1 , X1, Y1, g1) such that(AS1 , S1) is in F , X1 is a subset of S1 with n − 1 non-spherical indecomposable components, Y1 is a subset of S1,and g1 is in AS1 . Let X1 be a non-spherical indecomposable component of X . We have g AX1 g−1

⊆ AY . Then, byhypothesis, g = hg1 with h ∈ AY and g1 ∈ AX⊥

1. Now in the subgroup AX⊥

1we have g1 AX−X1 g−1

1 ⊆ AY∩X⊥

1. Since

F is an SBSPS-family, we get, by induction hypothesis, that g1 = h′g2 with h′∈ AY and g2 ∈ A(X−X1)

⊥as

∩ AX⊥

1. But

A(X−X1)⊥as

∩ AX⊥

1= AX⊥

as. Thus g = hh′g2 with hh′

∈ AY and g2 ∈ AX⊥as

. �

Lemma 4.6 (Second Restriction Lemma). Let F be an SBSPS-family. Assume that for every (AS, S) in F , every twosubsets X and Y of S, with X indecomposable with at most one element s such that X − {s} is both indecomposableand not of spherical type and every g in AS , (Ď) is true; then (Ď) is true for every (AS, S) of F , every two subsets Xand Y of S and every g in AS .

Proof. Assume that (Ď) is true for every (AS, X, Y, g), such that (AS, S) is in F , g is in AS , and X and Y are subsetsof S such that X is indecomposable with at most one element s such that X − {s} is both indecomposable and not ofspherical type. By the First Restriction Lemma, it is enough to prove that (Ď) is true when X is indecomposable andnot of spherical type. Let (AS, S) be in F , let X and Y be in S. Assume that s and t are two distinct elements of Xwith the property that X1 = X − {s} and X2 = X − {t} are indecomposable and not of spherical type. Let g be in ASsuch that g AX g−1 is a subgroup of AY . Since g AX1 g−1 is then a subgroup of AY we have g = hg1 with h in AY andg1 in AX⊥

1. Since F is an SBSPS-family, A{t}⊥∪X2

is in F . As t is in X1, the set X⊥

1 is a subset of {t}⊥ and g1 is in

A{t}⊥∪X2. Furthermore, in A{t}⊥∪X2

we have the inclusion g1 AX2 g−11 ⊆ AY∩({t}⊥∪X2)

.Thus g1 = h′g2 with h′ in AY and g2 in A{t}⊥∪X2

∩ AX⊥

2= AX⊥ . �

4.2. Projections on Deligne subcomplexes

We prove Theorem 4.8 concerning the existence of rectangles in CAT(0) spaces and apply it in Corollary 4.9 toCAT(0) Deligne complexes. We first need a technical lemma.

50 E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52

Lemma 4.7. Let (D, d) be a CAT(0) metric space and G a group that acts by isometries on D. Let C1 and C2 be twonon-empty convex subspaces of D and let x be in C1. Assume that Fix(x) ⊆ Stab(C1) ∩ Stab(C2) and that for everyy in C1 different from x, the subgroup Fix(x) is not a subgroup of Fix(y); then we have πC1(πC2(x)) = x.

Proof. Applying Lemma 3.6 to x and C = C2, we get the inclusion Fix(x) ⊆ Fix(πC2(x)). Applying again thislemma to πC2(x) and C = C1, we get that the subgroup Fix(x) ⊆ Fix(πC1(πC2(x))). By the second hypothesisapplied to y = πC1(πC2(x)) we get x = πC1(πC2(x)). �

Theorem 4.8 (Rectangle Theorem). Let (D, d) be a CAT(0) metric space and G a group that acts by isometries onD. Let C1 and C2 be two non-empty convex subspaces of D. Assume that x and y are two distinct points of C1 not inC2 and such that

(1) Fix(x) ⊆ Stab(C1) ∩ Stab(C2);

(2) ∀z ∈ C1, Fix(x) ⊆ Fix(z) ⇐⇒ x = z;(3) Fix(y) ⊆ Stab(C1) ∩ Stab(C2);

(4) ∀z ∈ C1, Fix(y) ⊆ Fix(z) ⇐⇒ y = z.

Then [x, πC2(x), πC2(y), y] is isometric to a rectangle of the Euclidean plane E2.

Note that the angles 6 x (πC2(x), y), 6(πC2 (x))(x, πC2(y)), 6

(πC2 (y))(πC2(x), y) and 6 y(πC2(y), x) are consequently allequal to π

2 ; the projection on C2 of the mid-point of x and y is the mid-point of πC2(x) and πC2(y); and we have theequality d(x, y) = d(πC2(x), πC2(y)).

Proof. By the previous lemma we have x = πC1(πC2(x)) and y = πC1(πC2(y)). Thus by Proposition 2.4, we get thefour inequalities 6

πC2 (y)(πC2(x), y) ≥π2 ; 6

πC2 (x)(x, πC2(y)) ≥π2 ; 6 x (πC2(x), y) ≥

π2 and 6 y(πC2(y), x) ≥

π2 .

Hence, by Proposition 2.5, [x, πC2(x), πC2(y), y] is isometric to a convex quadrilateral of E2 which must be arectangle. �

Corollary 4.9. Let (AS, S) be an Artin–Tits system such that the Moussong realization ΓS of its Deligne complex DSis CAT(0). Let X, Y be subsets of S and g ∈ AS such that g AX g−1

⊆ AY . Assume that X is not of spherical typeand that AX1 and AX2 are two distinct maximal standard parabolic subgroups of spherical type of AX such that g isnot in AY . Then,

[AX1 , πg−1·ΓY(AX1), πg−1·ΓY

(AX2), AX2 ]

is isometric to a rectangle of the Euclidean plane E2.

Proof. In the notation of Theorem 4.8, set C1 = ΓX and C2 = g−1· ΓY , x = e · AX1 and y = e · AX2 . Note

that Stab(C1) = AX and Stab(C2) = g−1 AY g; it is clear that AX is included in Stab(C1). Considering the imageof e · A∅ by g ∈ Fix(C) we get that g is in AX . We have Fix(e · AX1) ∪ Fix(e · AX2) = AX1 ∪ AX2 ⊆ AX ⊆

AX ∩ g−1 AY g = Stab(C1) ∩ Stab(C2). Furthermore hypotheses (2) and (4) of Theorem 4.8 are a consequence ofthe maximality assumption. Note that neither AX1 or AX2 are in g−1

· ΓY , since g is not in AY . Hence we can applyTheorem 4.8. �

4.3. The case of two-dimensional Artin–Tits groups

Let us recall that (AS, S) is a two-dimensional Artin–Tits system if every standard parabolic subgroup AX ofspherical type satisfies #X ≤ 2. For this section, we fix a two-dimensional Artin–Tits system (AS, S). We identifythe Deligne complex ΓS with its Moussong realization. We fix X, Y in S and g in AS such that g AX g−1

⊆ AY .Our objective, in this section, is to prove (Ď) under these hypotheses. By the first and second restriction lemmas,namely Lemmas 4.5 and 4.6, we can assume that X is indecomposable with at most one s such that X − {s} isindecomposable and not of spherical type. In our particular case, this implies that #X ≤ 3 with at most one pair {s, t}such that ms,t = ∞. If X contains a maximal standard parabolic subgroup of spherical type, then by Corollary 3.10,the element g is in AY are we are done. So we assume that X does not contain such a subgroup. With the previousrestrictions and the two-dimensional assumption, this means that X = {s, t} with ms,t = ∞.

If u is in S, we write, for short, Au for A{u}.

E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52 51

Proposition 4.10. Let (AS, S) be a two-dimensional Artin–Tits system. Let s, t be in S such that ms,t = ∞; setX = {s, t}. Let Y be a subset of S and g in AS such that g AX g−1 is included in AY . Then g = uv with v ∈ AX⊥ andu ∈ AY . As a consequence, X is a subset of Y .

In order to prove this proposition we use the following lemma.

Lemma 4.11. Let u, v be in S such that mu,v is finite, that is {u, v} ∈ S f,S . Then, we have 6 e·A∅(e·Au, e·Av) ∈ [

π2 ; π [

in the Moussong realization of the Deligne complex ΓS . Furthermore, we have 6 e·A∅(e · Au, e · Av) =

π2 if and only

if mu,v = 2.

Proof. By construction K (e · A∅, e · A{u,v}) is a quadrilateral of E2 with two right angles at the vertices Au andAv . But the angle 6 e·A{u,v}

(e · Au, e · Av) is acute; thus the angle 6 e·A∅(e · Au, e · Av) is in [

π2 ; π [. Furthermore,

6 e·A∅(e · Au, e · Av) =

π2 if and only if 6 e·A{u,v}

(e · Au, e · Av) =π2 , that is mu,v = 2. �

Proof of Proposition 4.10. We prove the result using an induction on the number n of transversal points of thegeodesic [A∅, πg−1·ΓY

(A∅)]. Note that A∅ is a transversal point and that πg−1·ΓY(A∅) is also a transversal point by

Lemma 3.9.The main idea is to study the rectangle [As, πg−1·ΓY

(As), πg−1·ΓY(At ), At ]. Set Fix(πg−1·ΓY

(A∅)) = g−1h AZ withh ∈ AY and Z ⊆ Y .

If n = 1, then A∅ = πg−1·ΓY(A∅) = g−1h AZ . Thus g is in AY and we are done. If n = 2, then span(A∅, πg−1·ΓY

(A∅))

exists and g−1h is in AZ ; again g is in AY . So assume n ≥ 3 and that g is not in AY . Also, assume that for every hin AS , such that h AX h−1 is included in AY and the number of transversal points of the geodesic [A∅, πh−1·ΓY

(A∅)] islower than n, we can write h = uv with v ∈ AX⊥ and u in AY . Since g is not in AY , by the same arguments as in thebeginning of the proof, both geodesics [As, πg−1·ΓY

(As)] and [At , πg−1·ΓY(At )] have at least three transversal points.

Denote by γ(1)s the first transversal point of [As, πg−1·ΓY

(As)] distinct from As and set span(γ(1)s ) = K (h AR, h AT );

we have Fix(γ(1)s ) = h AR . By definition of γ

(1)s , span(As, γ

(1)s ) exists. By Lemma 2.2, it follows that As ∩ h AR 6= ∅.

Therefore, h AR = s j AR and K (h AR, h AT ) = K (s j AR, s j AT ) for some j ∈ Z. But the subgroup As fixes the pointse · As and πg−1·ΓY

(e · As); then it fixes each point of the geodesic joining them and, in particular, it fixes γ(1)s , that is

As is a subgroup of s j ARs− j . Hence Fix(γ(1)s ) = AR and s is in R. Since the point γ

(1)s is transversal, R cannot be

equal to {s}; since AS is a two-dimensional Artin–Tits group, γ(1)s = e · A{s,u} for some u ∈ S such that ms,u 6= ∞.

Denote by γ(1)t the first transversal point of [e · At , πg−1·ΓY

(e · At )] distinct from e · At . Similarly, we obtain

γ(1)t = e · A{t,v} for some v in S such that mt,v 6= ∞. We are going to prove now that u = v, that mu,s = mu,t = 2,

and that u is in X⊥.The geodesic triangle [e · A∅, e · As, e · A{s,u}] is included in the geodesic rectangle [e · As, πg−1·ΓY

(e ·

As), πg−1·ΓY(e · At ), e · At ]. Thus by construction of DS and ΓS , it follows that B(e · A∅, ε) ∩ K{s,u} is a subset

of [e · As, πg−1·ΓY(As), πg−1·ΓY

(e · At ), e · At ] for some positive ε. In the same way there exists a positive ε′ such thatB(A∅, ε

′) ∩ K{t,v} is a subset of the rectangle [As, πg−1·ΓY(As), πg−1·ΓY

(At ), At ]. Since K{s,u} ∩ K{t,v} = {e · A∅}, itfollows that

0 ≤ 6 e·A∅(e · As, e · Au) + 6 e·A∅

(e · At , e · Av) ≤ 6 e·A∅(e · As, e · At ) = π.

52 E. Godelle / Journal of Pure and Applied Algebra 208 (2007) 39–52

But, by the previous lemma, both angles are at least equal to π2 . Hence, we have 6 e·A∅

(e · As, e · Au) = 6 e·A∅(e · At , e ·

Av) =π2 , that is ms,u = mt,v = 2. Also, K{s,u} and K{t,v} are rectangles. Therefore u = v and the first transversal

point of [e · A∅, πg−1·ΓY(e · A∅)] distinct from e · A∅ is γ

(1)∅

= Au , the mid-point of e · A{u,s} and e · A{u,t}. Consider

now γ(2)s , γ

(2)t and γ

(2)∅

the respective third transversal points of [e · As, πg−1·ΓY(e · As)], [e · At , πg−1·ΓY

(e · At )] and

[e · A∅, πg−1·ΓY(e · A∅)]. The point γ

(1)s , that is e · A{s,u}, and the point γ

(2)s span a cell, and γ

(2)s is fixed by As . Thus,

by arguments similar to the ones used to prove that γ(1)s = e · A{s,u}, and that ms,u = 2, we get γ

(2)s = u j As for some

j ∈ Z − {0}. Furthermore, u j· K{s,u} and u j

· K{t,u} are rectangles, then we have γ(2)t = u j At and γ

(2)∅

= u j A∅.

Now consider g1 = gu j . Since u ∈ AX⊥ , we have g1 AX g−11 = g AX g−1

⊆ AY . But, AS acts by isometries on ΓS

and u− j send u j As , u j At , u j A∅, u j·ΓX and g−1

·ΓY to e · As , e · At , e · A∅, ΓX and g−11 ·ΓY respectively. Hence, the

number of transversal points of [e · A∅, πg−11 ·ΓY

(e · A∅)] is n − 2 and we can apply the induction hypothesis: g1 = uv1

with u ∈ AY and v1 ∈ AX⊥ . Finally g = uv with v = v1u− j∈ AX⊥ . �

Corollary 4.12. Let (AS, S) be a two-dimensional Artin–Tits system. The group AS has Property (~).

As a corollary, we obtain Theorem 3.Conjecture 4.2 remains open, but we expect that the method developed here can be extended to every Artin–Tits

group, at least under Conjecture 2.6.

Acknowledgments

I am grateful to John Crisp and Luis Paris for fruitful discussions. I also thank Patrick Dehornoy for his helping tomake the paper more reader-friendly.

References

[1] J. Altobelli, R. Charney, A geometric rational form for Artin groups of FC type, Geom. Dedicata 79 (2000) 277–289.[2] N. Bourbaki, Groupes et Algebres de Lie chapitres 4,5,6, Hermann, 1968.[3] M. Bridson, A. Haefliger, Metric Spaces of Non Positive Curvature, Springer-Verlag, 1999.[4] E. Brieskorn, K. Saito, Artin–Tits Gruppen und Coxeter Gruppen, Invent. Math. 17 (1972) 245–271.[5] R. Charney, The Tits conjecture for locally reducible Artin groups, Int. J. Algebra Comput. 10 (2000) 783–797.[6] R. Charney, The Deligne complex for the four-strand braid group, Trans. Amer. Math. Soc. 356 (2004) 3881–3897.[7] R. Charney, M. Davis, The K (π, 1)-problem for hyperplane complements associated with infinite reflection groups, J. Amer. Math. Soc. 8

(1995) 597–627.[8] P. Deligne, Les immeubles des groupes de tresses generalises, Invent. Math. 17 (1972) 273–302.[9] R. Fenn, D. Rolfsen, J. Zhu, Centralisers in the braid group and singular braid monoid, Enseign. Math. 42 (1996) 75–96.

[10] E. Godelle, Normalisateur et groupe d’Artin de type spherique, J. Algebra 269 (2003) 263–274.[11] E. Godelle, Parabolic subgroups of Artin groups of type FC, Pacific J. Math 208 (2003) 243–254.[12] G. Moussong, Hyperbolic Coxeter groups, Ph.D. Thesis, Ohio State University, 1988.[13] L. Paris, Parabolic subgroups of Artin groups, J. Algebra 196 (1997) 369–399.[14] L. Paris, Artin monoids inject in their groups, Comment. Math. Helv. 77 (2002) 609–637.[15] D. Rolfsen, Braid subgroup normalisers and commensurators and induced representations, Invent. Math. 130 (1997) 575–587.[16] H. Van der Lek, The homotopy type of complex hyperplane complements, Ph.D. Thesis, Nijmegen, 1993.